3824 OPTICS LETTERS / Vol. 34, No. 24 / December 15, 2009

All-optical tunable pulse frequency chirpvia slow light

Gil Fanjoux* and Thibaut SylvestreInstitut FEMTO-ST, Département d’Optique P. M. Duffieux, Université de Franche-Comté,

CNRS UMR 6174, Besançon, France*Corresponding author: gil.fanjoux@univ-fcomte.fr

Received July 17, 2009; revised September 9, 2009; accepted September 18, 2009;posted November 9, 2009 (Doc. ID 114439); published December 8, 2009

We theoretically investigate slow light via stimulated Raman scattering, paying special attention to the pi-cosecond regime where chromatic dispersion and cross-phase modulation must be considered. In addition tothe control of the Raman pulse walk-off, we demonstrate that the cross-phase-modulation-induced frequencychirp can also be all-optically tuned via Raman slow light. We further demonstrate that this new implicationis a consequence of the fact that the group velocity is significantly more affected than the phase velocity inslow-light media. © 2009 Optical Society of America

OCIS codes: 190.0190, 190.5650, 190.5890, 190.3270, 260.2030, 230.2035.

Slow and fast light (SFL) in nonlinear media has at-tracted considerable interest for several years andhas been observed in plenty of configurations [1].SFL generally requires a medium exhibiting sharpresonance with steep dispersion, which in turn re-sults in a strong change of group velocity at the reso-nance, leading to optical delay or advancement. SFLcan be achieved by using a medium with a frequency-dependent absorption or gain or by using othermechanisms such as atomic resonances with narrowelectromagnetically induced transparency (EIT) win-dows, coherent population oscillations (CPO), andlaser-induced amplifying resonances [1]. For the de-velopment of fiber-based tunable delay lines at tele-communication wavelengths [2], SFL has also beenstudied in optical fibers with stimulated Brillouinscattering (SBS) [3,4] or stimulated Raman scatter-ing (SRS) [5] or by using narrowband fiber paramet-ric amplification [6].

The aim of this Letter is to stress that SFL affectslight propagation in nonlinear material well beyondoptical delay or advancement. SFL can indeed impactother physical parameters than group velocity, and,in particular, the pulse frequency chirp. In this Let-ter, we theoretically demonstrate in a slow-light Ra-man medium, where both group-velocity dispersion(GVD) and cross-phase modulation (XPM) must beconsidered, that the XPM-induced frequency shiftcan be all-optically controlled by tuning the pulsewalk-off via slow light. Specifically, this is achievedby considering picosecond pulses propagating in anarrowband Raman fiber amplifier system that al-lows for the achievement of slow-light optical delaysin the picosecond timescale. This could be imple-mented, for example, in a highly nonlinear hollow-core photonic crystal fiber [7] filled with carbon disul-fide �CS2�, which is characterized by a strong andnarrowband Raman gain enabling picosecond opticaldelays [8,9].

Let us start our analysis by recalling the usualtheory of the frequency chirp induced by a pumppulse on a small signal pulse at the Stokes Raman

frequency through the combined effects of XPM and

0146-9592/09/243824-3/$15.00 ©

GVD in an optical fiber. For normal GVD, the Stokessignal pulse predominantly interacts with the pumpleading edge because of GVD-induced pulse walk-off,yielding an XPM-induced down-frequency shift. As-suming pump and signal Gaussian pulses synchro-nized at the medium input, the total frequency chirpacross the signal pulse can be expressed at the wave-guide output and in the signal reference frame as[10,11]

����� = −�PLw

�T0�exp�− �2� − exp�− �� − �w�2��, �1�

with � the nonlinear coefficient, P the input pumppower, T0 the pulse width, �= �t−z /vgs� /T0 the nor-malized time, and �w=L /Lw with L the fiber length.Lw is the walk-off length expressed as Lw=T0�1/vgp−1/vgs�−1=LT0��tL�−1 with vgp, and vgs the group ve-locity of the pump and the signal, respectively, with�tL=L�R�2 the GVD temporal walk-off, where �2 isthe GVD coefficient and ��R /2�� the Raman fre-quency detuning between the pump and the signalpulses [8]. The XPM-induced frequency shift at thesignal pulse peak is then obtained by setting �=0 inEq. (1),

���0� = −�PLw

�T0�1 − exp�− �w

2 ��. �2�

Equation (2) yields a down-frequency shift linearlydependent on the pump power [10], as shown by thesolid thin line in Fig. 1. This usual situation is alsoillustrated in the numerical simulation depicted inFig. 2(a) that shows the normalized pump and signalpulses as well as the frequency chirp function, re-spectively. The inset of Fig. 2(a) shows the signalspectrum at the medium input and output, respec-tively, together with the associated down-frequencyshift and the Raman gain band. The numerical simu-lation is based on the nonlinear Schrödinger equation(NLSE) including the Raman susceptibility, and theparameters are listed in the caption of Fig. 2. The

real part of the Raman susceptibility has been delib-

2009 Optical Society of America

December 15, 2009 / Vol. 34, No. 24 / OPTICS LETTERS 3825

erately removed from the simulation so as not to in-clude the slow-light effect [9].

If we now consider the slow-light process Eq. (2) isno longer valid, because slow light generates an ad-ditional tunable pulse walk-off and therefore leads tothe full control of the signal pulse peak position withrespect to both that of the pump pulse and of theXPM-induced chirp function. However the chirp func-tion of Eq. (1) remains valid, because the slow lightdoes not significantly affect the signal phase in ourconditions, i.e., in a narrowband Raman amplifier inwhich the change of the group index is significantlylarger than that of the phase index. It is indeed well-known that the group velocity is much more affectedby slow light than the phase velocity, since the formerparameter is the derivative of the latter [1].

To take into account the new temporal position ofthe signal peak in the same reference frame as before(i.e., signal without slow light), we must define a nor-malized slow-light length as LSL=T0�1/vgs

SL−1/vgs�−1

=LT0��tSL�−1 where vgsSL is the slow group velocity of

the signal pulse. �tSL= �gRPL����RAeff�−1 is the opti-cal delay generated by Raman slow light, with���R /2�� the gain bandwidth, gR the Raman gain,and Aeff the effective core area of the fiber [8]. Be-cause of the slow-light optical delay, the temporal po-sition of the signal envelope is now equal to �=L /LSL=�SL and not �=0, as in the previous casewithout slow light. The resulting XPM-induced fre-quency shift in the slow-light regime is thus given by

����SL� = −�PLw

�T0 �exp�− �SL

2 � − exp�− ��SL − �w�2��.

�3�

Compared to Eq. (2), Eq. (3) reveals that one canhave full control of both the sign and the magnitudeof the XPM-induced frequency chirp simply by tuningthe pump power because of its new dependence on�SL. This behavior is illustrated in Fig. 1, whichshows the total pulse walk-off �t=�tL−�tSL (dashed

Fig. 1. Pulse walk-off (left, dashed) and XPM-induced fre-quency chirp (right, solid) as a function of the pump powerP. Thin curve, standard theory without slow light, Eq. (2).Gray curve, analytical results with slow light [Eq. (4)].Black curve, NLSE numerical results.

gray curve) and the associated XPM-induced fre-

quency chirp (solid gray curve), respectively, as afunction of the pump power. Our analytical resultshave also been checked against numerical simula-tions (black curves in Fig. 1) that show the same ten-dencies for the walk-off and the frequency chirp, re-spectively. Figure 1 shows that the slow-light effectresults in a pulse walk-off that decreases with thepump power, whereas in the absence it was constant,as recently demonstrated in [8]. More importantly, italso results on a frequency chirp that is predomi-nantly positive, whereas in the absence it was nega-tive. In particular, Fig. 1 reveals two interesting re-gimes. The first one concerns the removal of theRaman pulse walk-off by slow light when �t=�tL−�tSL=0, obtained for the critical pump power Pcr=�2�R��RAeffgR

−1 [8]. In such conditions, Eq. (3) pre-dicts that the frequency shift is unexpectedly nonzero

Fig. 2. (Color online) Frequency chirp ����� (left, dottedcurve) and pump and signal pulses profiles (dashedand solid normalized curves, respectively). The parametersare T0=250 ps, �=4.134 m−1.W−1, Lw=4.69 m, L=2 m,��R /2��=20 THz, ���R /2��=15 GHz, Aeff=10 m2, �2=4.2410−25 s2 m−1, and gR=23.210−11 m W−1 at thepump wavelength of 532 nm. (a) Without slow light for P=Pcr, similar to Eq. (1). (b) Zero walk-off for P=Pcr=0.285 W. (c) Zero frequency shift for P=0.17 W. The in-sets show the signal spectrum at the medium’s input (dot-ted curves) and output (solid curves), respectively, and theRaman gain (dashed curves).

and equal to [by setting �SL=�w in Eq. (3)]

3826 OPTICS LETTERS / Vol. 34, No. 24 / December 15, 2009

����w� = −�PLw

�T0�exp�− �w

2 � − 1�, �4�

yielding an up-frequency shift, which is of the orderof 0.9 GHz for the analytical value. Surprisingly, it isthe exact opposite value of the down-frequency shiftpredicted by Eq. (2), i.e., in the absence of slow light.(See solid thin curve in Fig. 1.) This is counterintui-tive: one may reasonably expect zero frequency shift,since there is no longer any walk-off between signaland pump pulses. Slow light exactly compensates forthe GVD, leading to group-velocity matching betweenthe two pulses. However, as this group velocitymatching is not due to the absence of dispersion butto slow light, this nonzero frequency shift can be in-terpreted as resulting from the major difference be-tween the phase and the group velocity variations ina slow-light medium. In such a case, we have to aban-don the classical picture of the XPM theory where thechirp function exactly matches the pump intensitydistribution. This is due to the fact that the pulse en-velope is slowing down, whereas the chirp function isnot affected by slow light, because it relies on thephase variation. Correspondingly, the signal phasepropagates faster than the signal envelope and there-fore travels across it. This results in a signal phasethat is predominantly influenced by the pump trail-ing edge, leading to an up-frequency shift. This inter-pretation is verified by the numerical simulation. Asshown in Fig. 2(b), even if the pulse walk-off is fullycanceled by slow light, the signal still undergoes anup-frequency shift as shown in the inset of Fig. 2(b),as if the signal pulse predominantly interacted withthe pump trailing edge. It can also be viewed in Fig.2(b) that the zero point of the chirp function is stillobtained for �=�w /2 as for Fig. 2(a), which clearlymeans that the chirp function is not affected by slowlight.

The second case is even more intriguing and corre-sponds to the removal of the XPM-induced frequencyshift at the pulse peak, as shown by the solid curvesin Fig. 1. Equation (3) indeed predicts that the fre-quency shift can be set to zero when LSL=2Lw, i.e.,vgs

SL= �vgp+vgs� /2, which corresponds to an inputpump power that is half of the critical pump powerP=Pcr /2 as shown in Fig. 1. Note that this also im-plies that the chirp is less important than for boththe previous cases at P=Pcr. As can be seen in Fig.2(c), the signal peak is now at the same temporal po-sition as the zero point of the chirp function, whereasthe pulse walk-off is no longer canceled ��t=30 ps�.The inset of Fig. 2(c) shows that the signal spectrumno longer suffers from any frequency shift, despitethe fact that the signal pulse interacts with the lead-ing edge of the pump pulse. Note that factor 2 be-tween the discussed regimes, i.e., zero walk-off andzero frequency chirp, is simply related to the fact thatthe zero point of the chirp function is set to �=�w /2,as predicted by Eq. (1). For pump power larger thanPcr, the pulse walk-off becomes negative due to theslowing down of the signal pulse, leading to an in-

duced frequency shift that is predominantly positiveas if it propagates in the anomalous dispersion re-gime.

In addition, the small discrepancies between ana-lytical and numerical results in Fig. 1 rely mainly onthe fact that the analytical expression of the Ramanslow light delay is valid only in the steady-state re-gime, i.e., it assumes that the pump is a cw field andthat the signal bandwidth is smaller than the Ramangain bandwidth [9]. As the signal center frequencycan shift out of the gain bandwidth due to XPM, thisleads to the decrease of the slowing-down of the sig-nal pulse [see the insets of Figs. 2(a) and 2(b)]. It ex-plains the fact that the numerical critical pumppower needed to cancel the walk-off is higher thanthe analytical value. This is also consistent with thefact that the numerically calculated XPM-inducedfrequency shift is slightly greater than the analyticalvalues.

In conclusion, we have theoretically demonstratedthat in an optical fiber system with a narrowbandRaman gain, not only the pulse walk-off but also thepulse frequency chirp can be all-optically tunedNegative or positive frequency shift can be achievedin this way simply by tuning the pulse walk-off usingslow light. We have shown that this new implicationis a direct consequence of the fact that the phase ve-locity is not affected by slow light compared to thegroup velocity. Note that we expect that this phenom-enon cannot be observed in silica fibers nor with SBS,because of its backward nature, nor with SRS due toits broad gain bandwidth that leads to femtosecondoptical delays much lower than the GVD delays [5].

The authors thank M. Gonzàlez-Herràez and L.Thévenaz for helpful discussions, and S. Coen forcareful reading of the manuscript. This work is par-tially funded by the European INTERREG IV-A pro-gramme and COST action 299, and by the ConseilRégional de Franche-Comté.

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